I took a hardship withdrawal from my 401k in 2022 to purchase my first home. This should exempt the withdrawal from the 10% penalty when I file my taxes, correct?

The exception to the 10% early withdrawal penalty for a first time home purchase is only if the funds were distributed from an IRA, not a 401(k).

IRS website for IRA exceptions - https://www.irs.gov/taxtopics/tc557

IRS website for non IRA exceptions - https://www.irs.gov/taxtopics/tc558

No.   You are not exempt from the 10% early withdrawal penalty when you take money out of a 401k.  That only works for a traditional IRA.   When you take money out of a 401k for a down payment you are subject to not only the ordinary income tax but the 10% early withdrawal penalty.  You misunderstood the rules.

Sorry. Even though they have a similar purpose, IRAs and 401(k) are controlled by different tax laws and follow different rules.  The homebuyer exception does not apply to a 401(k).